Consider the system of differential equations Find the general solution to this system, given that the general solution to the corresponding homo- geneous system is xh(t) = C₁e²t x₁ (t) = 3x₁ (t) + 2x₂(t) x₂(t) = x1(t) + x₂(t) + e²t x₂ (1)= ae²¹ Given the system of differential equations x₁' (1)=3x₁ (1)+2x₂ (1) x₂ (1) = -x₁ (1) + x₂ (1) e²¹ and the homogeneous solution is sint-cost x₁ (1)=C₁e²¹ cost The objective is to find the general solution to the system. (sint-cost) + C₂e²t (- ^) ( +C₂e²¹ Take derivative. x,' (1)-(20₁) e²¹ Plug the values into the given system. a₁ (3) -(39) + (0) 29₂ Find the particular solution. By using the method of undetermined coefficients, the particular solution is of the form = (a ₁)e²+ 21 3a,e² +2a₂e² -a₁e²1 + a₂e²¹ + e²² (sin! The particular solution is xp The general solution is x(1) = x₁ (1) + x₂ (1) =G₁e² Equating the coefficients. 2a₁ = 3a₁ +2a₂ a₁ +2a₂ = 0 2a₂ = a₁ + a₂ +1 a₁ + a₂ = 1 Solving both equations gives a₁ = 2 and a₂ = -1. - cost - sin sin t -cost-sint sint sint-cost COS/ sint) 2 =(²₁) ₁². e²1 2 J + C₂e³² ( - Sini) + (²1) ² -cost-sint sint elp with this one ind the unique solution to the inhomogeneous system above which satisfies x(0) = =
Consider the system of differential equations Find the general solution to this system, given that the general solution to the corresponding homo- geneous system is xh(t) = C₁e²t x₁ (t) = 3x₁ (t) + 2x₂(t) x₂(t) = x1(t) + x₂(t) + e²t x₂ (1)= ae²¹ Given the system of differential equations x₁' (1)=3x₁ (1)+2x₂ (1) x₂ (1) = -x₁ (1) + x₂ (1) e²¹ and the homogeneous solution is sint-cost x₁ (1)=C₁e²¹ cost The objective is to find the general solution to the system. (sint-cost) + C₂e²t (- ^) ( +C₂e²¹ Take derivative. x,' (1)-(20₁) e²¹ Plug the values into the given system. a₁ (3) -(39) + (0) 29₂ Find the particular solution. By using the method of undetermined coefficients, the particular solution is of the form = (a ₁)e²+ 21 3a,e² +2a₂e² -a₁e²1 + a₂e²¹ + e²² (sin! The particular solution is xp The general solution is x(1) = x₁ (1) + x₂ (1) =G₁e² Equating the coefficients. 2a₁ = 3a₁ +2a₂ a₁ +2a₂ = 0 2a₂ = a₁ + a₂ +1 a₁ + a₂ = 1 Solving both equations gives a₁ = 2 and a₂ = -1. - cost - sin sin t -cost-sint sint sint-cost COS/ sint) 2 =(²₁) ₁². e²1 2 J + C₂e³² ( - Sini) + (²1) ² -cost-sint sint elp with this one ind the unique solution to the inhomogeneous system above which satisfies x(0) = =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Find the unique solution to the inhomogeneous system above which satisfies x(0) = matrix ....
Transcribed Image Text:Consider the system of differential equations
Find the general solution to this system, given that the general solution to the corresponding homo-
geneous system is
x₁ (t) = 3x₁ (t) + 2x₂(t)
x₂ (t) = x1(t) + x₂(t) + e²t
xh(t) = C₁e²t (sin
x₂ (1)= ae²¹
= (a ₁) ²²
Given the system of differential equations
x' (1)=3x₁ (1)+2x₂ (1)
x₂² (1) = -x₂₁ (1) + x₂ (1) e²¹
and the homogeneous solution is
sint-cost
-cost-sint
x₂ (1)=C₁e²¹
+C₂e²¹
cost
sint
The objective is to find the general solution to the system.
Find the particular solution. By using the method of undetermined coefficients,
the particular solution is of the form
Take derivative.
x,' (1)=(201₁) e²¹
Plug the values into the given system.
3
2
a₁
(29)² =(³, 1)(a)~²¹ +(2₂)
-1
3a,e² +2a₂e²
-a₁e²² + a₂e²¹ +e²₁
→
t - cost
cos t
Equating the coefficients.
2a₁ = 3a₁ +2a₂
2a₂ = -a₁ + a₂ +1
a₁ + a₂ = 1
Solving both equations gives a, = 2 and a₂ =-1.
st) + C₂e²1 (-
+
The general solution is
x(1) = x₂ (1) + x₂ (1)
=Ge²
The particular solution is x,
q₁ +2a₂ = 0
sint-cost
cost
2
--(²)
- cost - sin t
sin t
sint)
051) + C₂0²³ ( 1)+(²1) ²
-cost-sint
sint
Help with this one
Find the unique solution to the inhomogeneous system above which satisfies x(0)
-
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